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Exam 1 Solutions February 3, 2000 ECE 221: Electric Circuits Dr. McNames • Write your 6-digit identification number and student identification numbers below. • Do not begin the exam or look at the problems until instructed to do so. • You have 100 minutes to complete the exam. • Once you begin, write your student ID at the top of each page. • Do not use separate scratch paper. If you need more space, use the backs of the exam pages. • If you have extra time, double check your answers. If you run out of time, write a note describing your strategy and equations that can be used to help solve the problem. Problem 1:______ / Problem 2:______ / Problem 3:______ / Problem 4:______ / 17 20 13 10 Total:______ / 60 6-Digit Identification Number:_____________ Student Identification Number:_____________ Student ID:_________________ 2 of 6 1. Mesh Current Method (17 points) 10 Ω ia 2v 1 50 Ω 20 V ic 60 Ω ib 80 Ω 40 Ω v1 a. (9 pts) For the circuit shown above, write the mesh-current equations around each loop in terms of the currents ia , ib , and ic. You do not need to simplify your equations. Loop ia : 10i a + 2 ⋅ 40i c + 50(i a − i c ) = 0 Loop ib : − 2 ⋅ 40i c + 60ib + 80(ib − ic ) = 0 Loop ic : 50(i c − ia ) + 80( ic − ib ) + 40ic = 20 b. (3 pts) Solve for the currents ia , ib , and ic. ia = -96.6 mA ib = 221 mA ic = 193 mA c. (2 pts) How much power is being delivered by the independent source? P20V = 20 ic = 3.86 W d. (2 pts) How much power is being delivered by the dependent source? P2V1 = 2 ⋅ v1 (ib − ia ) = 2 ⋅ 40ic ( ib − i a ) = 4.90 W e. (1 pt) How much power is being absorbed by the resistors? PR = P20V + P2V1 = 8.76 W Student ID:_________________ 3 of 6 2. Node Voltage Method (20 pts) 30iα i14 4Ω 1 50Ω 2 6Ω 2Ω 3 iα v1 20 V 40 Ω v3 20 Ω 4 v4 a. (6 pts) For the circuit shown above, write the node-voltage equations at each of the specified nodes in terms of v 1 , v3 , v 4 , and i14 . You do not need to simplify your equations. Node 1: v1 v1 − 20 + + i14 = 0 50 4 Node 3: v3 − 20 v3 v3 − v4 + + =0 6 40 2 Node 4: v4 − v3 v 4 + − i14 = 0 2 20 b. (1 pt) Write an expression for iα in terms of v 3 and v 4 . iα= v4 − v3 2 c. (2 pts) In part a. you should have written 3 independent equations in 4 unknowns. Write a fourth independent equation that relates the voltages between nodes 1 and 4 to the voltages v 3 and v 4 . Hint: use your answer from part b. Eq: 30( v4 − v3 ) = v4 − v1 2 Student ID:_________________ 4 of 6 2. Node Voltage Method – Continued The same circuit from the previous page is shown below for convenience. 30iα i14 4Ω 1 50Ω 2 6Ω 2Ω 3 iα v1 20 V 40 Ω v3 20 Ω 4 v4 d. (6 pts) Use a supernode to write three independent equations in terms of v 1 , v3 , and v 4 . You do not need to simplify your equations. Eq. 1: v1 v1 − 20 v4 − v3 v 4 + + + =0 50 4 2 20 Eq. 2: v3 − 20 v3 v3 − v4 + + =0 6 40 2 Eq. 3: 30( v4 − v3 ) = v4 − v1 2 e. (3 pts) Solve for the voltages v 1 , v3 , and v 4 . v1 = 14.7 V v3 = 18.0 V v4 = 18.2 V f. (2 pts) How much power is being delivered by the dependent source? v4 − v3 =115 mA 2 v P30 = 30iα ⋅ i14 = 30iα iα + 4 =3.54 W 20 iα = Student ID:_________________ 5 of 6 3. Thevenin/Norton Equivalents & Maximum Power Transfer (13 pts) Hint: think carefully about how to approach this problem. Some approaches are easier than others. a b 8 kΩ 20 kΩ 40V 30 kΩ 2 mA 5 kΩ a. (4 pts) Find the current flowing from the node labeled a to the node labeled b if the nodes are connected together (short circuit current). Solve part b. first, then calculate Isc by the relationship of Req to Voc. Req = (20 k || 30k ) + 8k = 20 kΩ ISC = VOC = -1.2 mA Req b. (4 pts) Find the voltage between the terminals a and b if the nodes are left unconnected (open circuit voltage), as shown. Label the voltage across the 30k Ohm resistor as V and apply the node-voltage method. V − 40 V + − 2m = 0 ; V = 48 V 20 k 30k VOC= 40 − V − 2m ⋅ 8k = -24 V c. (4 pts) Draw the Thevenin and Norton equivalents of the circuit as seen from the nodes a and b. Clearly label these nodes. 20 kΩ a 24 V a 1.2 mA b 20 kΩ b d. (1 pt) Suppose a resistor RL is connected to the nodes a and b. What value of RL will maximize the power delivered to RL ? RL = 20 kΩ Student ID:_________________ 6 of 6 4. Operational Amplifiers (10 pts) For both circuits below, assume that the operational amplifier is ideal and operating in the linear region. a. (4 pts) Find the values of Rf and Rb so that v o = 20ib – 10ia . Rf Ra ia vo RL Rb ib v p = ib Rb vo − v p Rf = −i a vo = Rbi b − R f ia Rb = 20 Ω Rf = 10 Ω b. (6 pts) Find an expression for v o as a function of v 1 and i2 . R3 R1 R2 v1 R4 vf i2 v f = − R4 i2 vf v −v − R4 i2 = o 1 = R3 R3 + R1 R3 R + R1 R4i 2 vo = v1 − 3 R3 RL vo